Strongly summable ultrafilters on N and small maximal subgroups of ßN.
Supercompactness and failures of GCH
Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show...
Supercompactness and partial level by level equivalence between strong compactness and strongness
We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if is regular, δ is strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.
Template iterations and maximal cofinitary groups
Jörg Brendle (2003) used Hechler’s forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that , the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial...
The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.
Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...
The Amalgamation Property, the Universal-Homogeneous Models, and the Generic Models.
The Arkhangel’skiĭ–Tall problem: a consistent counterexample
We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
The Arkhangel'skiĭ–Tall problem under Martin’s Axiom
We show that MA implies that normal locally compact metacompact spaces are paracompact, and that MA() implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
The axiom of choice and linearly ordered sets
The axiom of choice for linearly ordered families
The cardinal equation 2m=m
The combinatorics of reasonable ultrafilters
We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
The consistency of 𝔟 = κ and 𝔰 = κ⁺
Using finite support iteration of ccc partial orders we provide a model of 𝔟 = κ < 𝔰 = κ⁺ for κ an arbitrary regular, uncountable cardinal.
The consistency of some theorems concerning Lebesgue measure (Preliminary communication)
The consistency of the measurability of projective semisets
The consistency strength of the tree property at the double successor of a measurable cardina
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the...
The distributivity numbers of finite products of P(ω)/fin
Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o., is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).
The dual group of a dense subgroup
Throughout this abstract, is a topological Abelian group and is the space of continuous homomorphisms from into the circle group in the compact-open topology. A dense subgroup of is said to determine if the (necessarily continuous) surjective isomorphism given by is a homeomorphism, and is determined if each dense subgroup of determines . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...
The enriched stable core and the relative rigidity of HOD
In the author's 2012 paper, the V-definable Stable Core 𝕊 = (L[S],S) was introduced. It was shown that V is generic over 𝕊 (for 𝕊-definable dense classes), each V-definable club contains an 𝕊-definable club, and the same holds with 𝕊 replaced by (HOD,S), where HOD denotes Gödel's inner model of hereditarily ordinal-definable sets. In the present article we extend this to models of class theory by introducing the V-definable Enriched Stable Core 𝕊* = (L[S*],S*). As an application we obtain...
The equiconsistency of two large cardinal axioms